Behavior of Early Warnings near the Critical Temperature in the Two-Dimensional Ising Model

Abstract

Among the properties that are common to complex systems, the presence of critical thresholds in the dynamics of the system is one of the most important. Recently, there has been interest in the universalities that occur in the behavior of systems near critical points. These universal properties make it possible to estimate how far a system is from a critical threshold. Several early-warning signals have been reported in time series representing systems near catastrophic shifts. The proper understanding of these early-warnings may allow the prediction and perhaps control of these dramatic shifts in a wide variety of systems. In this paper we analyze this universal behavior for a system that is a paradigm of phase transitions, the Ising model. We study the behavior of the early-warning signals and the way the temporal correlations of the system increase when the system is near the critical point.

Fig 1. Spatial configurations in the Ising model.

Typical spatial configurations for a 2-dimensional Ising model. Three regimes are shown: a) T < T _c, b) T ≈ T _c and c) T > T _c. Black squares represent spins with σ = +1 and white one correspond to σ = −1.

Fig 2. Total magnetization as a function of time in the Ising model.

Typical behavior of the total magnetization time series in a 2-dimensional Ising model. Three regimes are shown: a) T < T _c, b) T ≈ T _c and c) T > T _c. It is important to notice the change of scale between plots.

Fig 3. Temporal mean as a function of temperature.

Ensemble behavior of the mean as a function of temperature. The mean corresponds to the total magnetization of the system. Three regimes are shown, T < T _c, T ≈ T _c and T > T _c. Note that the mean can also approach −1 at low temperatures; we only show here the positive values.

Fig 4. Temporal variance as a function of temperature.

Ensemble behavior of the variance as a function of temperature. Three regimes are shown, T < T _c, T ≈ T _c and T > T _c.

Fig 5. Absolute values of temporal skewness as a function of temperature.

Ensemble behavior of the skewness as a function of temperature. Three regimes are shown, T < T _c, T ≈ T _c and T > T _c.

Fig 6. Temporal kurtosis as a function of temperature.

Ensemble behavior of the kurtosis as a function of temperature. Three regimes are shown, T < T _c, T ≈ T _c and T > T _c.

Fig 7. Temporal auto correlation at lag 1 as a function of temperature.

Ensemble behavior of the autocorrelation function for lag τ = 1 as a function of temperature. Three regimes are shown, T < T _c, T ≈ T _c and T > T _c.

Fig 8. Power Spectral Density as a function of temperature.

Ensemble behavior of the Power Spectral Density as a function of temperature. Panel (a) shows the behavior of the PSD for temperatures T ≤ T _c. Temperature increases from bottom to top, with T _c corresponding to the topmost curve. Panel (b) shows the behavior of the PSD for temperatures T ≥ T _c. Temperature increases from top to bottom, with T _c corresponding to the topmost curve. The crossover frequency for each temperature is shown as a red dot.

Fig 9. Crossover frequency as a function of temperature.

Behavior of the crossover frequency as a function of temperature. Three regimes are shown, T < T _c, T ≈ T _c and T > T _c.